Answer:
Step-by-step explanation:To show that multiplying polynomials is a closed system, we'll multiply the given polynomials and demonstrate that the result is still a polynomial.
Given polynomials:
�
(
�
)
=
�
2
−
1
P(x)=x
2
−1
�
(
�
)
=
−
2
�
+
5
Q(x)=−2x+5
Multiply
�
(
�
)
P(x) by
�
(
�
)
Q(x):
(
�
2
−
1
)
(
−
2
�
+
5
)
(x
2
−1)(−2x+5)
Expanding this:
=
�
2
×
(
−
2
�
)
+
�
2
×
5
−
1
×
(
−
2
�
)
−
1
×
5
x
2
×(−2x)+x
2
×5−1×(−2x)−1×5
=
−
2
�
3
+
5
�
2
+
2
�
−
5
−2x
3
+5x
2
+2x−5
The result,
−
2
�
3
+
5
�
2
+
2
�
−
5
−2x
3
+5x
2
+2x−5, is also a polynomial. Thus, the product of two polynomials is still a polynomial, demonstrating that polynomial multiplication is a closed operation.